Question #140549
Q.1 examine the convergence of series: 3×4/5^2 +'5×6/7^2 + 7×8/9^2 + ......
1
Expert's answer
2020-10-27T19:06:39-0400

3.452+5.672+7.892+=n=2(2n1).(2n)(2n+1)2=n=2Un\frac{3.4}{5^2}+\frac{5.6}{7^2}+\frac{7.8}{9^2}+\dots = \sum_{n=2}^\infin\frac{(2n-1).(2n)}{(2n+1)^2}= \sum_{n=2}^\infin U_n

Then we have

limnUn=limn(2n1).(2n)(2n+1)2=limn(21n).2(2+1n)2=2.24=44=10\lim_{n\to \infin} U_n= \lim_{n\to\infin}\frac{(2n-1).(2n)}{(2n+1)^2} = \lim_{n\to\infin}\frac{(2-\frac{1}{n}).2}{(2+\frac{1}{n})^2}= \frac{2.2}{4}=\frac{4}{4}=1 \ne0

Since limnUn0\lim_{n\to \infin}U_n\ne0 then by Cauchy's test for divergence the series is divergent(not convergent)


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